Absence of collapse into persistent Dirac-type singularities in a Keller-Segel-Navier-Stokes system involving local sensing

The chemotaxis-Navier-Stokes system $$ \begin{cases} n_t+u\cdot\nabla n = \Delta \big(n\phi(c)\big),\\ c_t+ u\cdot\nabla c = \Delta c -c+n,\\ u_t + (u\cdot\nabla) u =\Delta u+\nabla P + n\nabla\Phi, \qquad \nabla\cdot u=0, \end{cases} $$ is considered along with homogeneous boundary conditions of no-flux type for $n$ and $c$, and of Dirichlet type for $u$, in a smoothly bounded planar domain $\Omega$, where $\Phi \in W^{2,\infty}(\Omega)$ is given. A concept of generalized solvability is introduced, and in this framework a statement on global existence is derived for all reasonably regular initial data $(n_0,c_0,u_0)$, provided that $\phi\in C^3((0,\infty))$ is positive and bounded on $(\xi_0,\infty)$ for each $\xi_0>0$, and such that $$ \xi\phi(\xi) \to +\infty \qquad \mbox{and} \qquad \frac{\xi\phi'^2(\xi)}{\phi(\xi)} \to 0 \qquad \mbox{as } \xi\to\infty. $$ These solutions particularly satisfy $\int_0^T \Omega n\ln (n+1) < \infty$ for all $T > 0$ as well as $\int_\Omega n(\cdot,t)=\int_\Omega n_0$ for a.e. $t > 0$.